#9206642

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(11,14)(16,18)(19,21)(23,24)(25,26,31,29)(27,34,33,30)(28,36,38,32)(35,37,39,40);(1,2,5,4)(3,6,8,7)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(26,29)(27,33)(32,36)(35,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,40)(38,39);(3,8)(6,7)(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,26)(27,34,33,30)(28,36,38,32)(35,40,39,37);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30,31,34)(26,33,29,27)(28,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113838);